To this end, we view our -approximate solution at time level , as an (N+1)-dimensional column vector which is uniquely realized
at the Gauss collocation nodes .

The forward Euler time-differencing (meth_cheb.7a)
with homogeneous boundary conditions
(meth_cheb.7b), reads
where L is an matrix which accounts for the
spatial spectral differencing together with the homogeneous boundary
conditions,

Theorem 4.1 tells us that if the CFL condition
(meth_cheb.12) holds, i.e., if
then is bounded in the -weighted induced
operator norm,

Let us consider an (s + 2)-level time differencing method of the form
In this case, is given by a
convex combination of stable forward Euler differencing, and we
conclude

. Assume
that the following CFL condition holds,Then the spectral approximation (cheb_RK.4)
is strongly stable, and the
following estimate holds

Second and third-order accurate multi-level time differencing methods
of the positive type (cheb_RK.4)
take the particularly
simple form
with positive coefficients, , given in Table
4.1

Table 4.2:

Similar arguments apply for Runge-Kutta time-differencing methods. In this
case the resulting positive type Runge-Kutta methods take the form